1. Slide 15-1 Copyright © 2004 Pearson Education, Inc.

2. Slide 15-2 Copyright © 2004 Pearson Education, Inc. Probability Rules! Chapter 15 Created by Jackie Miller, The Ohio State University

3. Slide 15-3 Copyright © 2004 Pearson Education, Inc. Recall That… For any random phenomenon, each trial generates an outcome. An event is any set or collection of outcomes. The collection of all possible outcomes is called the sample space, denoted S.

4. Slide 15-4 Copyright © 2004 Pearson Education, Inc. Events When outcomes are equally likely, probabilities for events are easy to find just by counting. When the k possible outcomes are equally likely, each has a probability of 1/k. For any event A that is made up of equally likely outcomes,.

5. Slide 15-5 Copyright © 2004 Pearson Education, Inc. The First Three Rules for Working with Probability Rules Make a picture.

6. Slide 15-6 Copyright © 2004 Pearson Education, Inc. Picturing Probabilities The most common kind of picture to make is called a Venn diagram. We will see Venn diagrams in practice shortly…

7. Slide 15-7 Copyright © 2004 Pearson Education, Inc. The General Addition Rule When two events A and B are disjoint, we can use the addition rule for disjoint events from Chapter 14: P(A or B) = P(A) + P(B) However, when our events are not disjoint, this earlier addition rule will double count the probability of both A and B occurring. Thus, we need the general addition rule.

8. Slide 15-8 Copyright © 2004 Pearson Education, Inc. The General Addition Rule (cont.) General Addition Rule: –For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B) The following Venn diagram shows a situation in which we would use the general addition rule:

9. Slide 15-9 Copyright © 2004 Pearson Education, Inc. It Depends… Back in Chapter 3, we looked at contingency tables and talked about conditional distributions. Now we will talk about conditional probabilities. A probability that takes into account a given condition is called a conditional probability.

10. Slide 15-10 Copyright © 2004 Pearson Education, Inc. It Depends… (cont.) To find the probability of the event B given the event A, we restrict our attention to the outcomes in A. We then find the fraction of those outcomes B that also occurred. Formally,. Note: P(A) cannot equal 0, since we know that A has occurred.

11. Slide 15-11 Copyright © 2004 Pearson Education, Inc. The General Multiplication Rule When two events A and B are independent, we can use the multiplication rule for independent events from Chapter 14: P(A and B) = P(A) x P(B) However, when our events are not independent, this earlier multiplication rule does not work. Thus, we need the general multiplication rule.

12. Slide 15-12 Copyright © 2004 Pearson Education, Inc. The General Multiplication Rule (cont.) We encountered the general multiplication rule in the form of conditional probability. Rearranging the equation in the definition for conditional probability, we get the general multiplication rule: –For any two events A and B, P(A and B) = P(A) x P(B|A) or P(A and B) = P(B) x P(A|B)

13. Slide 15-13 Copyright © 2004 Pearson Education, Inc. Independence Independence of two events means that the outcome of one event does not influence the probability of the other. With our new notation for conditional probabilities, we can now formalize this definition: –Events A and B are independent whenever P(B|A) = P(B). (Equivalently, events A and B are independent whenever P(A|B) = P(A).)

14. Slide 15-14 Copyright © 2004 Pearson Education, Inc. Independent ≠ Disjoint Disjoint events cannot be independent! Well, why not? –Since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn’t. –Thus, the probability of the second occurring changed based on our knowledge that the first occurred. –It follows, then, that the two events are not independent.

15. Slide 15-15 Copyright © 2004 Pearson Education, Inc. Depending on Independence It’s much easier to think about independent events than to deal with conditional probabilities. –It seems that most people’s natural intuition for probabilities breaks down when it comes to conditional probabilities. Don’t fall into this trap: whenever you see probabilities multiplied together, stop and ask whether you think they are really independent.

16. Slide 15-16 Copyright © 2004 Pearson Education, Inc. Drawing Without Replacement Sampling without replacement means that once one individual is drawn it doesn’t go back into the pool. –We often sample without replacement, which doesn’t matter too much when we are dealing with a large population. –However, when drawing from a small population, we need to take note and adjust probabilities accordingly. Drawing without replacement is just another instance of working with conditional probabilities.

17. Slide 15-17 Copyright © 2004 Pearson Education, Inc. Tree Diagrams A tree diagram helps us think through conditional probabilities by showing sequences of events as paths that look like branches of a tree. Making a tree diagram for situations with conditional probabilities is consistent with our “make a picture” mantra.

18. Slide 15-18 Copyright © 2004 Pearson Education, Inc. Tree Diagrams (cont.) Figure 15.4 is a nice example of a tree diagram and shows how we multiply the probabilities of the branches together:

19. Slide 15-19 Copyright © 2004 Pearson Education, Inc. Reversing the Conditioning Reversing the conditioning of two events is rarely intuitive. Suppose we want to know P(A|B), but we know only P(A), P(B), and P(B|A). We also know P(A and B), since P(A and B) = P(A) x P(B|A) From this information, we can find P(A|B):

20. Slide 15-20 Copyright © 2004 Pearson Education, Inc. Reversing the Conditioning (cont.) When we reverse the probability from the conditional probability that you’re originally given, you are actually using Bayes’s Rule.

21. Slide 15-21 Copyright © 2004 Pearson Education, Inc. What Can Go Wrong? Don’t use a simple probability rule where a general rule is appropriate: –Don’t assume that two events are independent or disjoint without checking that they are. Don’t find probabilities for samples drawn without replacement as if they had been drawn with replacement. Don’t reverse conditioning naively.

22. Slide 15-22 Copyright © 2004 Pearson Education, Inc. Key Concepts The addition rule for disjoint events can be generalized for any two events using the general addition rule. The multiplication rule for independent events can be generalized for any two events using the general multiplication rule. Conditional probabilities come from the general multiplication rule. Tree diagrams are helpful ways to display conditional events and probabilities.